Certainty Equivalent and Exponential Utility

The buying price of a project can be computed easily if it is assumed that the investor's utility function is of exponential form, U (.v) = — e~ox for some a > 0 The computing procedure uses certainty equivalents rather than expected values,

Let us briefly review the certainty equivalent concept Suppose that an investor has a utility function U, Suppose that X is a random variable describing the investor's wealth at the terminal point. Then the expected utility of this wealth is E[U (X)]. The certainty equivalent is the (nonrandom) amount J such that U(x) = E[U(X)]. We often write CE(X) for the certainty equivalent of X

As a specific case suppose that U(X) = — e~"x and suppose that the random variable X has two possible outcomes Xi and X2 occurring with probabilities p\ and p2, respectively. The expected utility is

To find the certainty equivalent ~x we solve e~a* = Ple-"Xl + p2e~"X2

Taking the logarithm of both sides, we obtain

CE(X) =x ^ --\r\{P]e-"Xi 4- pie~,lX2}, (16 11)

This may look complicated, but it has a very special and important property

The special property of this form is that if a constant, say A, is added to a random variable, the certainty equivalent increases by this same constant, This property is often referred to as the delta property. Formally,

for any random variable X and any constant A This property can be checked easily for the two-outcome case by referencing (1611).

Here is a general proof for exponential utility. We have

Therefore,

This says that

CE(X + A) = CE(X) + A . This delta property only holds for utility functions that are exponential or linear,

Delta property A utility function is linear or exponential if and only if for all random ~~y variables X and all constants A, the certainty equivalent satisfies

Ci FIGURE 16,10 Simple project. This project has initial cash flow c<>, followed at the end of the period by a cash flow of value either o or Cj

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